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Biological Fluid Flow Project 07 Development of a Numerical Model to Describe Steady Streaming in Unsteady Axisymmetric Biological Flows in a Geometry with a Slowly Varying Cross-Section Laboratory of Computational Engineering Dr. Mark Sawley Laboratory of Fluid Mechanics Prof. Peter Monkewitz - Radboud Nelissen Development of a Numerical Model to Describe Steady Streaming in Unsteady Axisymmetric Biological Flows in a Geometry with a Slowly Varying Cross-Section Laboratory of Computational Engineering Dr. Mark Sawley Laboratory of Fluid Mechanics Prof. Peter Monkewitz - Radboud Nelissen Anthony Claude Abstract Cerebrospinal fluid moves from the cranial cavities all along the cerebrospinal canal in a pulsatile way. Its flow has been studied here through various geometries. From the basic case of a tube to a model of an idealized anatomical case, velocity profile and flow behaviour were plotted, analyzed and compared. This bi-dimensional study takes into account variation of geometry of the cerebrospinal canal and pulsatile character of the flow. A data base relating variation of internal and external wall for an adult has been realized with Visual Human Server1. This research allows determining Reynolds and Womersley numbers average for a given region of the spinal canal. Quantitative solutions of unsteady analysis through a cylindrical tube and qualitative results concerning steady streaming in geometries with slowly variations of section have been found . Finally comparison were carried out between computational calculations made in 1998 by A. Sarkar and G. Jayaraman2 and Fluent calculations for a similar issue. 1http://visiblehuman.epfl.ch/vhve.php 2Correction to flow rate - pressure drop relation in coronary angioplasty: steady straeming effect Contents I Basic Fluid Mechanics Applied to Biological Flows 4 II Womersley Number’s Influence on Biological Flows 5 1 Introduction 5 2 Pulsatile flow through a tube 6 2.1 Hypotheses . 6 2.2 Velocity’s expression . 6 2.3 Results’ Interpretation . 9 2.4 RMS-velocity’s expression . 11 3 Pulsatile flow through an annular tube 12 3.1 Velocity’s expression . 12 3.2 Results’ Interpretation . 13 III Numerical Flow Modelling 14 4 Introduction 14 5 Steady state flow - analytical vs. numerical results 15 5.1 Analytical solution . 15 5.2 Numerical solution . 18 5.3 Results’ interpretation . 20 6 CSF flow through an axisymmetric annular model 21 6.1 Generating the mesh . 22 6.2 Results’ interpretation . 23 7 CSF flow through 2-D models with varying cross-sections 33 7.1 Visual Human Server - Database . 33 7.2 Theoritecal part . 35 7.3 Axisymmetric geometry with a linear variation of section . 37 7.4 Axisymmetric geometry with a half cosinusoidal variation of section . 39 7.5 Axisymmetric geometry with a cosinusoidal variation of geometry . 40 7.6 Sarkar and Jayaraman case . 43 8 Conclusion 46 9 Bibliography 47 Introduction Biofliudics has gained in importance in recent years, forcing engineers to use fluid me- chanic and apply it to biological situation. Cardiovascular and more generally every human flow circulation problems are very important nowadays. Because of the multiple diseases caused by flow comportment, it is fundamental to understand its characteristics and the rules by which they operate.. Why does an artery get stenosed? How could we optimized drug dispersion in the cerebrospinal canal? What is the influence of a cardiac valve on the blood flow? By answering these questions, we could offer better treatments and solutions to patients. At the EPFL, some experimental researches are conducted on the spread of drugs ad- ministered directly in the spinal canal (with a catheter). Furthermore, this project will also focus on cerebrospinal fluid flow. Its aim is to evaluate the potential of Fluent solver to calculate accurate solution for oscillatory flow. With the development of numerical simulation possibilities and the increase of preci- sion and liability of computer programs, research progresses faster. But technology progress is not enough, we still need an engineer to check, analyse and interpret results. That is why good knowledge in fluid mechanics is needed before solving a problem numerically. To what extent is it possible to consider the results obtained as an exact solution? Is it possible to control the precision, and if so, in what way? This project follows this path. Unfortunately the spinal canal geometry is extremely complex and varies between peo- ple. Considering that and the calculation time needed, it is not appropriate to model precisely the cerebrospinal canal. In order to study flow behaviour of these geometries with varying cross-section, an axisymmetric analytical model will be used. 4 Part I Basic Fluid Mechanics Applied to Biological Flows As every incompressible fluid, biological one obeys the different conservation principles. Traditionally we consider the conservation of the mass, the momentum and the energy: dρ + ρ∇ · v = 0 (1) dt Considering the incompressibility condition (ρ = cte) we obtain: ∇ · v = 0 (2) The conservation of the momentum using the Gauss theorem (local form) is ∂ (ρv) + ∇ · (ρvv) = ∇ · σ (3) ∂t Biological fluid flows are unsteady. Thus, they are driven by the gravitational force and the pressure gradient force (cardiac cycle governs this pulsatile flow). Some shear forces are also noticed(caused by the fluid’s viscosity).Many biological flow issue can be discribed analytically. For instance, recirculation caused by a stenosis in an artery. This phenomenon is important in hemodynamic because there is a risk of canal’s obstruction. Considering the cerebrospinal canal, we encounter the same problem. The difference is that the variation of geometry is due to the change of internal and external radius or to the nerve’s insertions and not to deposits.Most of the biological incompressible problems are solved using the Navier-Stokes equations with cylindrical coordinate. Navier-Stokes equation (in cylindrical coordinate) are: 2 ∂vr ∂vr vθ ∂vr ∂vr vθ • ρ ∂t + vr ∂r + r ∂θ + vz ∂z − r 2 2 2 ∂p ∂ vr 1 ∂ vr ∂ vr 1 ∂vr 2 ∂vθ vr = − ∂r + η ∂r2 + r2 ∂θ2 + ∂z2 + r ∂r − r2 ∂θ − r2 + ρfr ∂vθ ∂vθ vθ ∂vθ ∂vθ vθvr • ρ ∂t + vr ∂r + r ∂θ + vz ∂z + r 2 2 2 1 ∂p ∂ vθ 1 ∂ vθ ∂ vθ 1 ∂vθ 2 ∂vr vθ = − r ∂θ + η ∂r2 + r2 ∂θ2 + ∂z2 + r ∂r + r2 ∂θ − r2 + ρfθ ∂vz ∂vz vθ ∂vz ∂vz • ρ ∂t + vr ∂r + r ∂θ + vz ∂z 2 2 2 ∂p ∂ vz 1 ∂ vz ∂ vz 1 ∂vz = − ∂z + η ∂r2 + r2 ∂θ2 + ∂z2 + r ∂r + ρfz 5 Part II Womersley Number’s Influence on Biological Flows 1 Introduction Before undertaking more complex analysis of the cerebro-spinal canal, a study of an oscillatory fluid flow through a simple geometry is recommanded. The first part of this project is dedicated to an analysis of a flow moving by a pressure gradient (representing the cardiac cycle) in a cylindrical tube. It is crucial to understand the basic physics of the issue before developing a numeri- cal model. Therefore, the following development can be viewed as a theoretical approach of unsteady flows, particularly, of biological flows. The first step will be to solve Navier-Stokes equation in order to find an expression for the velocity. Once this is done, the profile shape can be plotted and observed. Finally, a discussion of the physical interpretation and of the parameters’ influence will take place. The aim here is to define the influence of the Womersley number on a fluid flow and its effect on the velocity profile. Cerebrospinal fluid - functions and properties The main function of the cerebrospinal fluid is to protect the brain. In fact, by bathing in this fluid, it does not feel effect of the pressure fluctuations. Obviously there are some limits and this protection is efficient as far as we are in a standard situation. Normally, CSF volume and blood volume vary inversely, maintaining the intracranial pressure within normal limits. CSF has also several other important functions: it regulates chemical environment of the central nervous system and it is active in the transportation of biological substances like proteins. It is principally composed by plasma with few proteins and cells and it is considered as a Newtonian flow g m2 Density ρ [ m3 ] Viscosity µ [ s ] CSF 993 0.0067 Table 1: CSF’s properties 2 PULSATILE FLOW THROUGH A TUBE 6 2 Pulsatile flow through a tube As a first approximation, the cerebrospinal canal will be modeled as a cylindrical tube. By doing so, the equations are simpler but further from reality. The main objective here is to study the influence of the Womersley number on a fluid flow and on its velocity profile. Figure 1: One-dimensional fluid flow in a tube 2.1 Hypotheses • Incompressible flow, no external forces ∂P iωt • Pressure grandient is periodic (sinus): ∂z = Ge , with G constant • One-dimensional fluid flow: (0, 0,W ) avec W = W (R, t) ∂ • Axisymmetric flow: ∂θ = 0 D • Boundary condition: W ( 2 , t) = 0 2.2 Velocity’s expression After some simplifications, Navier-Stokes equations become: ∂W ∂P 1 ∂ ∂W ρ = − + µ (R ) (4) ∂t ∂z R ∂R ∂R ∂P ∂P = = 0 (5) ∂R ∂θ ∂P = Geiωt (6) ∂z To introduce the Womersley number and to make the calculation easier, it is prefer- able to make this problem adimensional: τ t = ω R = R0r W = W0w 2 PULSATILE FLOW THROUGH A TUBE 7 q ω Womersley number α: α = R0 ν W0R0 Reynolds number Re: Re = ν ∂W 1 ∂P 1 ∂ ∂W = − + ν (R ) (7) ∂t ρ ∂z R ∂R ∂R ∂w 2 ∂p ν 1 ∂ ∂w ωW0 = −W0 + rW0 (8) ∂τ ∂z R0r R0 ∂r ∂r ∂w ∂p ∂2w 1 ∂w α2 = −ReR + + (9) ∂τ 0 ∂z ∂r2 r ∂r As mentioned aboved, the pressure gradient oscillates. It’s expression is: ∂p = Geiωt = Geiτ (10) ∂z The consequence of this is that the velocity will be oscillatory too: w(r, t) =w ˆ(r)eiωt =w ˆ(r)eiτ (11) To simplify the notation, wˆ(r, t) will be noted w hereafter.
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